Hecke operators are used to investigate part of the $\E_2$-term of
the Adams spectral sequence based on elliptic homology. The main
result is a derivation of $\Ext^1$ which combines use of classical
Hecke operators and $p$-adic Hecke operators due to Serre.

Let $G$ be a finite group, $H$ a copy of its $p$-Sylow
subgroup, and $\kn$ the $n$-th Morava $K$-theory at $p$.
In this paper we prove that the existence of an
isomorphism between $K(n)^\ast(BG)$ and $K(n)^\ast(BH)$ is
a sufficient condition for $G$ to be $p$-nilpotent.